Question

For what positive integers k is the following series convergent? \[\sum_{n=1}^{\infty}{\frac{(n!)^2}{(kn)!}}\]

Answer 1

looks like \[k\ge2\]

Answer 2

yes using the ratio test.

Answer 3

yes

Answer 4

how do you "simplify"
\[\frac{(kn)!}{(kn+k)!}\]? i know this goes to zero very rapidly, but can't figure out a nice form for it

Answer 5

\[\frac{1}{(kn+1)(kn+2)\cdots (kn+k)}\]

Answer 6

just 'stretch' out the factorial so it cancels

Answer 7

so you get
\[\frac{(n+1)^2}{(kn+1)(kn+2)\cdots (kn+k)}\] by ratio test or did i make a mistake?

Answer 8

that is correct